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					                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions




            Truthfulness and frugality ratio in the cheapest
                            path auctions
                                       Estonian Theory Days 2009


                                     Nick Gravin
                  (based on the unpublished joint paper with E. Elkind,
                                  N. Chen,F. Petrov)

                    St.Petersburg Department of Steklov Mathematical Institute RAS,

                                  Nanyang Technological University, Singapore


                                                   October 2009

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 General set system auctions. Model:


               There are given a set of agents E = {1, 2, . . . , n} and a family
               of feasible sets F ⊆ 2E .
               Auctioneer is intent on hiring a team (some feasible set) of
               agents — winning set.
               Each agent i has a true cost ci , but at the auction he can bid
               another price bi ≥ ci .
               Utility ui (profit) of an agent is bi − ci if he wins and 0 if he
               loses.
               Every agent is selfish and wants to maximize ui .



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 General set system auctions. Model:


               There are given a set of agents E = {1, 2, . . . , n} and a family
               of feasible sets F ⊆ 2E .
               Auctioneer is intent on hiring a team (some feasible set) of
               agents — winning set.
               Each agent i has a true cost ci , but at the auction he can bid
               another price bi ≥ ci .
               Utility ui (profit) of an agent is bi − ci if he wins and 0 if he
               loses.
               Every agent is selfish and wants to maximize ui .



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 General set system auctions. Model:


               There are given a set of agents E = {1, 2, . . . , n} and a family
               of feasible sets F ⊆ 2E .
               Auctioneer is intent on hiring a team (some feasible set) of
               agents — winning set.
               Each agent i has a true cost ci , but at the auction he can bid
               another price bi ≥ ci .
               Utility ui (profit) of an agent is bi − ci if he wins and 0 if he
               loses.
               Every agent is selfish and wants to maximize ui .



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 General set system auctions. Model:


               There are given a set of agents E = {1, 2, . . . , n} and a family
               of feasible sets F ⊆ 2E .
               Auctioneer is intent on hiring a team (some feasible set) of
               agents — winning set.
               Each agent i has a true cost ci , but at the auction he can bid
               another price bi ≥ ci .
               Utility ui (profit) of an agent is bi − ci if he wins and 0 if he
               loses.
               Every agent is selfish and wants to maximize ui .



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 General set system auctions. Model:


               There are given a set of agents E = {1, 2, . . . , n} and a family
               of feasible sets F ⊆ 2E .
               Auctioneer is intent on hiring a team (some feasible set) of
               agents — winning set.
               Each agent i has a true cost ci , but at the auction he can bid
               another price bi ≥ ci .
               Utility ui (profit) of an agent is bi − ci if he wins and 0 if he
               loses.
               Every agent is selfish and wants to maximize ui .



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 First price rule




       Auction is called a first price if auctioneer always chooses the
       cheapest feasible set, i.e. a set F with minimal e∈F b(e).




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 examples

       Non-trivial issues concerning computational complexity appears
       when feasible sets are given implicitly.
          1    Path auction: agents — edges in the graph; feasible sets —
               all paths between two given vertices s and t.
          2    k-Path: agents — edges in the graph; feasible sets — k
               edge-disjoint paths from s to t.
          3    Vertex cover auction: agents — vertices; feasible sets — all
               vertex covers of edges.
          4    Spanning tree auction: agents — edges; feasible sets —
               spanning trees.
          5    Perfect matching auction: agents — edges in bipartite
               graph; feasible sets — perfect matchings.

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 examples

       Non-trivial issues concerning computational complexity appears
       when feasible sets are given implicitly.
          1    Path auction: agents — edges in the graph; feasible sets —
               all paths between two given vertices s and t.
          2    k-Path: agents — edges in the graph; feasible sets — k
               edge-disjoint paths from s to t.
          3    Vertex cover auction: agents — vertices; feasible sets — all
               vertex covers of edges.
          4    Spanning tree auction: agents — edges; feasible sets —
               spanning trees.
          5    Perfect matching auction: agents — edges in bipartite
               graph; feasible sets — perfect matchings.

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 examples

       Non-trivial issues concerning computational complexity appears
       when feasible sets are given implicitly.
          1    Path auction: agents — edges in the graph; feasible sets —
               all paths between two given vertices s and t.
          2    k-Path: agents — edges in the graph; feasible sets — k
               edge-disjoint paths from s to t.
          3    Vertex cover auction: agents — vertices; feasible sets — all
               vertex covers of edges.
          4    Spanning tree auction: agents — edges; feasible sets —
               spanning trees.
          5    Perfect matching auction: agents — edges in bipartite
               graph; feasible sets — perfect matchings.

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 examples

       Non-trivial issues concerning computational complexity appears
       when feasible sets are given implicitly.
          1    Path auction: agents — edges in the graph; feasible sets —
               all paths between two given vertices s and t.
          2    k-Path: agents — edges in the graph; feasible sets — k
               edge-disjoint paths from s to t.
          3    Vertex cover auction: agents — vertices; feasible sets — all
               vertex covers of edges.
          4    Spanning tree auction: agents — edges; feasible sets —
               spanning trees.
          5    Perfect matching auction: agents — edges in bipartite
               graph; feasible sets — perfect matchings.

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 examples

       Non-trivial issues concerning computational complexity appears
       when feasible sets are given implicitly.
          1    Path auction: agents — edges in the graph; feasible sets —
               all paths between two given vertices s and t.
          2    k-Path: agents — edges in the graph; feasible sets — k
               edge-disjoint paths from s to t.
          3    Vertex cover auction: agents — vertices; feasible sets — all
               vertex covers of edges.
          4    Spanning tree auction: agents — edges; feasible sets —
               spanning trees.
          5    Perfect matching auction: agents — edges in bipartite
               graph; feasible sets — perfect matchings.

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 examples

       Non-trivial issues concerning computational complexity appears
       when feasible sets are given implicitly.
          1    Path auction: agents — edges in the graph; feasible sets —
               all paths between two given vertices s and t.
          2    k-Path: agents — edges in the graph; feasible sets — k
               edge-disjoint paths from s to t.
          3    Vertex cover auction: agents — vertices; feasible sets — all
               vertex covers of edges.
          4    Spanning tree auction: agents — edges; feasible sets —
               spanning trees.
          5    Perfect matching auction: agents — edges in bipartite
               graph; feasible sets — perfect matchings.

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Nash equilibrium in first price auction



       Definition
       Nash equilibrium(NE) in the first price auction is an assignment of
       agent bids b, such that no agent e can increase its utility ue by
       varying its bid be .

       NE exists if there is no agent belonging to all feasible
       sets(monopolist).
       We can find a NE explicitly in polynomial time.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Nash equilibrium in first price auction



       Definition
       Nash equilibrium(NE) in the first price auction is an assignment of
       agent bids b, such that no agent e can increase its utility ue by
       varying its bid be .

       NE exists if there is no agent belonging to all feasible
       sets(monopolist).
       We can find a NE explicitly in polynomial time.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Nash equilibrium in first price auction



       Definition
       Nash equilibrium(NE) in the first price auction is an assignment of
       agent bids b, such that no agent e can increase its utility ue by
       varying its bid be .

       NE exists if there is no agent belonging to all feasible
       sets(monopolist).
       We can find a NE explicitly in polynomial time.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions




       Truthful mechanisms




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Mechanisms for winners and payments selection


               Suppose the true costs of all agents are private.
               Agents have only one chance to send their bids to the
               auctioneer.
               Auctioneer (buyer, center) on the base of these bids should
               choose the winning set and how much to pay to them.

       Definition
       The mechanism of winners selection and their payments is called
       truthful if each agent maximizes utility by bidding its true cost
       (for any fixed bids of other agents).



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Mechanisms for winners and payments selection


               Suppose the true costs of all agents are private.
               Agents have only one chance to send their bids to the
               auctioneer.
               Auctioneer (buyer, center) on the base of these bids should
               choose the winning set and how much to pay to them.

       Definition
       The mechanism of winners selection and their payments is called
       truthful if each agent maximizes utility by bidding its true cost
       (for any fixed bids of other agents).



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Mechanisms for winners and payments selection


               Suppose the true costs of all agents are private.
               Agents have only one chance to send their bids to the
               auctioneer.
               Auctioneer (buyer, center) on the base of these bids should
               choose the winning set and how much to pay to them.

       Definition
       The mechanism of winners selection and their payments is called
       truthful if each agent maximizes utility by bidding its true cost
       (for any fixed bids of other agents).



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Mechanisms for winners and payments selection


               Suppose the true costs of all agents are private.
               Agents have only one chance to send their bids to the
               auctioneer.
               Auctioneer (buyer, center) on the base of these bids should
               choose the winning set and how much to pay to them.

       Definition
       The mechanism of winners selection and their payments is called
       truthful if each agent maximizes utility by bidding its true cost
       (for any fixed bids of other agents).



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Example




       (Vickrey) Second price auction (for particular set system).
       There are n sellers and auctioneer needs to bay exactly one thing.
       Mechanism selects the cheapest agent and pays to him the bid of
       the second by the price agent.

       Is it clear why this mechanism is truthful?




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Example




       (Vickrey) Second price auction (for particular set system).
       There are n sellers and auctioneer needs to bay exactly one thing.
       Mechanism selects the cheapest agent and pays to him the bid of
       the second by the price agent.

       Is it clear why this mechanism is truthful?




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Example




       (Vickrey-Clarke-Groves) VCG for general set system.
       Choose the cheapest feasible set F and pay independently to
       each agent in F its threshold bid, i.e. such bid that F is still the
       cheapest feasible set.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                  VCG
                                         Nash equilibriums
                                                                  frugality ratio
                                      Truthful mechanisms               √
                                                                  KKT       -mechanism
                        k-paths and cheapest path auctions


 VCG for cheapest path




                         c:                                   0                0
                                               0
                                   0                                                         0
                          S                                   1                                  T



       Initial true costs.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                  VCG
                                         Nash equilibriums
                                                                  frugality ratio
                                      Truthful mechanisms               √
                                                                  KKT       -mechanism
                        k-paths and cheapest path auctions


 VCG for cheapest path




                         c:                                   0
                                               0                             0
                                   0                                                       0
                          S                                   1                                  T



       Cheapest path.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                   VCG
                                         Nash equilibriums
                                                                   frugality ratio
                                      Truthful mechanisms                √
                                                                   KKT       -mechanism
                        k-paths and cheapest path auctions


 VCG for cheapest path




                         b:                                    1
                                                               5
                                               1                               1
                                                5                             5
                                  1                                                            1
                                      5                                                    5
                          S                                   1                                    T



       Cheapest Nash equilibrium. Total payment is 1.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                  VCG
                                         Nash equilibriums
                                                                  frugality ratio
                                      Truthful mechanisms               √
                                                                  KKT       -mechanism
                        k-paths and cheapest path auctions


 VCG for cheapest path




                         c:                                   0
                                               1                             0
                                   0                                                       0
                          S                                   1                                  T



       Payment to each agent on the winning path should be 1.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                  VCG
                                         Nash equilibriums
                                                                  frugality ratio
                                      Truthful mechanisms               √
                                                                  KKT       -mechanism
                        k-paths and cheapest path auctions


 VCG for cheapest path




                         c:                                   1
                                               1                             1
                                   1                                                       1
                          S                                   1                                  T



       The total payment is 5, but for NE it was only 1.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 How much mechanism overpays

               (E, F) — monopoly free set system;
               M — truthful mechanism for (E, F).
               c is a true cost vector; ν(c) — the total payment in the
               cheapest Nash for (E, F, c);
               PM (c) — total payment of M for c;

       Definition
       Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
       mechanism efficiency.

                                                                 PM (c)
                                              ΦM = sup                  .
                                                             c    ν(c)

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 How much mechanism overpays

               (E, F) — monopoly free set system;
               M — truthful mechanism for (E, F).
               c is a true cost vector; ν(c) — the total payment in the
               cheapest Nash for (E, F, c);
               PM (c) — total payment of M for c;

       Definition
       Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
       mechanism efficiency.

                                                                 PM (c)
                                              ΦM = sup                  .
                                                             c    ν(c)

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 How much mechanism overpays

               (E, F) — monopoly free set system;
               M — truthful mechanism for (E, F).
               c is a true cost vector; ν(c) — the total payment in the
               cheapest Nash for (E, F, c);
               PM (c) — total payment of M for c;

       Definition
       Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
       mechanism efficiency.

                                                                 PM (c)
                                              ΦM = sup                  .
                                                             c    ν(c)

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 How much mechanism overpays

               (E, F) — monopoly free set system;
               M — truthful mechanism for (E, F).
               c is a true cost vector; ν(c) — the total payment in the
               cheapest Nash for (E, F, c);
               PM (c) — total payment of M for c;

       Definition
       Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
       mechanism efficiency.

                                                                 PM (c)
                                              ΦM = sup                  .
                                                             c    ν(c)

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 How much mechanism overpays

               (E, F) — monopoly free set system;
               M — truthful mechanism for (E, F).
               c is a true cost vector; ν(c) — the total payment in the
               cheapest Nash for (E, F, c);
               PM (c) — total payment of M for c;

       Definition
       Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
       mechanism efficiency.

                                                                 PM (c)
                                              ΦM = sup                  .
                                                             c    ν(c)

                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Motivation for frugality ratio



               ΦM ≥ 1, since one can take c to be by itself a Nash
               equilibrium and then PM (c) ≥ ν(c).
               ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 )
               (E, F) is a Matroid iff for every two sets S, T ∈ F, there is a
               bijection f between S \ T and T \ S such that
               S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T .




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Motivation for frugality ratio



               ΦM ≥ 1, since one can take c to be by itself a Nash
               equilibrium and then PM (c) ≥ ν(c).
               ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 )
               (E, F) is a Matroid iff for every two sets S, T ∈ F, there is a
               bijection f between S \ T and T \ S such that
               S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T .




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Motivation for frugality ratio



               ΦM ≥ 1, since one can take c to be by itself a Nash
               equilibrium and then PM (c) ≥ ν(c).
               ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 )
               (E, F) is a Matroid iff for every two sets S, T ∈ F, there is a
               bijection f between S \ T and T \ S such that
               S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T .




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Mechanism design for cheapest path auctions


       Known results:
               VCG — ΦVCG for example with long path and an edge is
               quadratic in terms of the optimal frugality ratio.
                     √
               KKT      -mechanism gives a linear approximation to the
               optimal frugality ratio (Φ√ ≤ 2X and any truthful
                                                       1
               mechanism has frugality ratio at least √2 X ).
               Pruning-lifting mechanism (our paper) for k-path auctions
               provides optimal frugality ratio.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Mechanism design for cheapest path auctions


       Known results:
               VCG — ΦVCG for example with long path and an edge is
               quadratic in terms of the optimal frugality ratio.
                     √
               KKT      -mechanism gives a linear approximation to the
               optimal frugality ratio (Φ√ ≤ 2X and any truthful
                                                       1
               mechanism has frugality ratio at least √2 X ).
               Pruning-lifting mechanism (our paper) for k-path auctions
               provides optimal frugality ratio.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Mechanism design for cheapest path auctions


       Known results:
               VCG — ΦVCG for example with long path and an edge is
               quadratic in terms of the optimal frugality ratio.
                     √
               KKT      -mechanism gives a linear approximation to the
               optimal frugality ratio (Φ√ ≤ 2X and any truthful
                                                       1
               mechanism has frugality ratio at least √2 X ).
               Pruning-lifting mechanism (our paper) for k-path auctions
               provides optimal frugality ratio.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 KKT mechanism for cheapest path




       √
           -mechanism for cheapest path auctions.

       We will see that for some graphs it will have much better frugality
       ratio than VCG.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 First step.
       Find two edge-disjoint paths P, P minimizing b(P) + b(P ).
       (Ignore the rest of the graph)




                        S                                                                           T




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Second step.
       Let s = v1 , v2 , . . . , vk+1 = t be the vertices that P, P have in
       common, in the order in which they appear in P and P . Let Pi
       (resp. Pi ) be the subpath of P (resp. P ) from vi to vi+1 .




                   v1                                         v2
                                                                      v2                                 v3




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Third step.
       For each i, include Pi in the solution iff                              |Pi |b(Pi ) ≤           |Pi |b(Pi );
       otherwise, include Pi .



                   b:                                                                 0
                                                                          0                      3
                   v1                     5                    v2
                                                                       v2                                 v3
                         1                                 1
                                                                               0             4
                                          1


                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Forth step.
       Pay to each winner its threshold bid (i.e. the largest value that he
       can bid and still win, if the others bid the same).



                   b:                                                           4       2
                                                                                            −3




                                                                            3
                                                                                        3




                                                                         3 −
                                         √




                                                                                                 4
                                                                          2
                   v1                   3 3                    v2




                                                                                                 2
                                                                                                  3
                                                                        4
                                                                       v2                                 v3
                         1                                 1
                                                                               0             4
                                          1


                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Bounds on Φ√


               √
                   -mechanism is truthful, thus b = c.
               Φ√ ≤ 2 maxi                |Pi ||Pi |.
                                                                  √
               Thus for the edge and a path graph we have Φ√ = O(2 l),
               while ΦVCG = O(l).
                                                       1
               There is a lower bound                 √
                                                        2
                                                             maxi       |Pi ||Pi | on any truthful
               mechanism, thus
                              1
                             √ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
                               2 i                                                i




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Bounds on Φ√


               √
                   -mechanism is truthful, thus b = c.
               Φ√ ≤ 2 maxi                |Pi ||Pi |.
                                                                  √
               Thus for the edge and a path graph we have Φ√ = O(2 l),
               while ΦVCG = O(l).
                                                       1
               There is a lower bound                 √
                                                        2
                                                             maxi       |Pi ||Pi | on any truthful
               mechanism, thus
                              1
                             √ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
                               2 i                                                i




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Bounds on Φ√


               √
                   -mechanism is truthful, thus b = c.
               Φ√ ≤ 2 maxi                |Pi ||Pi |.
                                                                  √
               Thus for the edge and a path graph we have Φ√ = O(2 l),
               while ΦVCG = O(l).
                                                       1
               There is a lower bound                 √
                                                        2
                                                             maxi       |Pi ||Pi | on any truthful
               mechanism, thus
                              1
                             √ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
                               2 i                                                i




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 Bounds on Φ√


               √
                   -mechanism is truthful, thus b = c.
               Φ√ ≤ 2 maxi                |Pi ||Pi |.
                                                                  √
               Thus for the edge and a path graph we have Φ√ = O(2 l),
               while ΦVCG = O(l).
                                                       1
               There is a lower bound                 √
                                                        2
                                                             maxi       |Pi ||Pi | on any truthful
               mechanism, thus
                              1
                             √ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
                               2 i                                                i




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                                                 VCG
                                         Nash equilibriums
                                                                 frugality ratio
                                      Truthful mechanisms              √
                                                                 KKT       -mechanism
                        k-paths and cheapest path auctions


 k-paths and cheapest path auctions




       k-paths and cheapest path auctions




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Nash for k-paths


       Theorem
       Any Nash equilibrium for the cheapest path auction with respect
       to the bid vector b should have two non-intersecting by the edges
       cheapest paths.

       Theorem
       Any Nash equilibrium for the k-paths auction with respect to the
       bid vector b should have k + 1 non-intersecting by the edges
       cheapest paths.

       These theorems are non trivial results of graph theory.


                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Nash for k-paths


       Theorem
       Any Nash equilibrium for the cheapest path auction with respect
       to the bid vector b should have two non-intersecting by the edges
       cheapest paths.

       Theorem
       Any Nash equilibrium for the k-paths auction with respect to the
       bid vector b should have k + 1 non-intersecting by the edges
       cheapest paths.

       These theorems are non trivial results of graph theory.


                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Nash for k-paths


       Theorem
       Any Nash equilibrium for the cheapest path auction with respect
       to the bid vector b should have two non-intersecting by the edges
       cheapest paths.

       Theorem
       Any Nash equilibrium for the k-paths auction with respect to the
       bid vector b should have k + 1 non-intersecting by the edges
       cheapest paths.

       These theorems are non trivial results of graph theory.


                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Our mechanism for k-path auctions




       Pruning-lifting mechanism




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Pruning step



               Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that

                                         δ(P1 , . . . , Pk+1 ) = max b(P)
                                                                         P∈∪Pi

               is minimized. (Ignore the rest of the graph.)
               It is NP hard to find P1 , . . . , Pk+1 even with < k + 1 factor
               approximation of δ(P1 , . . . , Pk+1 ) to the optimal one.
               The cheapest k + 1 flow gives k + 1 approximation.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Pruning step



               Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that

                                         δ(P1 , . . . , Pk+1 ) = max b(P)
                                                                         P∈∪Pi

               is minimized. (Ignore the rest of the graph.)
               It is NP hard to find P1 , . . . , Pk+1 even with < k + 1 factor
               approximation of δ(P1 , . . . , Pk+1 ) to the optimal one.
               The cheapest k + 1 flow gives k + 1 approximation.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Pruning step



               Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that

                                         δ(P1 , . . . , Pk+1 ) = max b(P)
                                                                         P∈∪Pi

               is minimized. (Ignore the rest of the graph.)
               It is NP hard to find P1 , . . . , Pk+1 even with < k + 1 factor
               approximation of δ(P1 , . . . , Pk+1 ) to the optimal one.
               The cheapest k + 1 flow gives k + 1 approximation.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Lifting step




       We construct a graph H as follows:
               we take arcs of ∪i Pi as vertices for H;
               we draw an edge between e and e iff there is no path from s
               to t containing both e, e .




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Lifting step




       We construct a graph H as follows:
               we take arcs of ∪i Pi as vertices for H;
               we draw an edge between e and e iff there is no path from s
               to t containing both e, e .




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Lifting step




       We construct a graph H as follows:
               we take arcs of ∪i Pi as vertices for H;
               we draw an edge between e and e iff there is no path from s
               to t containing both e, e .




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Lifting step



         ∪Pi


          s                                                                                                         t




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Lifting step



         ∪Pi
                                     11
                                     00
                                     00
                                     11
                                     00
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                                                                                                      11
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                                                                                                      00
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                         11                                                                           11
                                                                                                      00
                         00
                         11                                                                           00
                                                                                                      11
                                                                                                      11
                                                                                                      00




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Lifting step



         H
                 1111111111111
                 000000000000011
                              00
                             000000
                             0000000000000000000000
                             111111
                             1111111111111111111111
                       00000000
                       11111111
                              11
                              00
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                 111111111111111
                              00                                                               00
                                                                                               11
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                 1111111111111                                                                 11
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                       00000000                                                                11
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                                                   11                                           0
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                  00000      000000
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                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Weighting step


               We split H into its components of connectivity H1 , . . . , Hl .
               Let Ai be an adjacency matrix of Hi .
               For each i we find the positive eigenvector (wi ) and
               eigenvalue αi of the matrix Ai . Let α = maxi αi .

                                                   (Ai ) (wi ) = αi (wi )
                                b(e)
               Define b (e) = w (e) . Let P the maximum weight path with
               respec to b (). Take ∪Pi \ P as the set of winners.
               Pay to each agent its threshold bid.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Weighting step


               We split H into its components of connectivity H1 , . . . , Hl .
               Let Ai be an adjacency matrix of Hi .
               For each i we find the positive eigenvector (wi ) and
               eigenvalue αi of the matrix Ai . Let α = maxi αi .

                                                   (Ai ) (wi ) = αi (wi )
                                b(e)
               Define b (e) = w (e) . Let P the maximum weight path with
               respec to b (). Take ∪Pi \ P as the set of winners.
               Pay to each agent its threshold bid.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Weighting step


               We split H into its components of connectivity H1 , . . . , Hl .
               Let Ai be an adjacency matrix of Hi .
               For each i we find the positive eigenvector (wi ) and
               eigenvalue αi of the matrix Ai . Let α = maxi αi .

                                                   (Ai ) (wi ) = αi (wi )
                                b(e)
               Define b (e) = w (e) . Let P the maximum weight path with
               respec to b (). Take ∪Pi \ P as the set of winners.
               Pay to each agent its threshold bid.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 Weighting step


               We split H into its components of connectivity H1 , . . . , Hl .
               Let Ai be an adjacency matrix of Hi .
               For each i we find the positive eigenvector (wi ) and
               eigenvalue αi of the matrix Ai . Let α = maxi αi .

                                                   (Ai ) (wi ) = αi (wi )
                                b(e)
               Define b (e) = w (e) . Let P the maximum weight path with
               respec to b (). Take ∪Pi \ P as the set of winners.
               Pay to each agent its threshold bid.



                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 ΦPL




                                          α
                                            ,         ΦPL =
                                          k
       where α is a lower bound on the frugality ratio of any truthful
             k
       mechanism.

       So PL is the optimal mechanism with respect to frugality ratio.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
                                          General auctions
                                         Nash equilibriums
                                      Truthful mechanisms
                        k-paths and cheapest path auctions


 ΦPL




                                          α
                                            ,         ΦPL =
                                          k
       where α is a lower bound on the frugality ratio of any truthful
             k
       mechanism.

       So PL is the optimal mechanism with respect to frugality ratio.




                                                                 Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)

				
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